258 APPENDIX TO CHAPTER IX

y1 into a road y2, such that q>l is defined in J x [0, 1] and ^>1(a, £) = y(a),
(^(/J, £) = y(/f) for any £. Define <p in I x [0, 1] as equal to q>± in J x [0, 1]
and to y(t) for any (t, £)e(I--J)x [0,1]; then for any x1eD1 (resp.
^c2 e D2), q> is a loophomotopy in C - -OJ (resp. C - {x2}) of y into a circuit
y3. We can therefore limit ourselves to proving that j(x^ y) — j(x2; y) = ± 1
when y is a circuit defined in I, having the following properties: (1) S c y(I)
and if T is the inverse image y"1^), then T is a subinterval of I and the
restriction of y to T is a homeomorphism of T onto S; (2) y(I — T) is contained
in C — D (note that perhaps this new y is not a simple loop). Then the inverse
image by y of the interval [—r, r] is a subinterval [A, /z] of T; suppose for
instance that y(l) = — r, y(/z) = r. We can suppose (replacing y by an equiva-
lent circuit) that A = — n, p, = 0, and moreover that — r is the origin of y,
so that I = [-TC, co] with co> 0. Take xl = /{, x2 = -/£ with 0 < { < r; let
a be the road t->• y(f), — TT ^ f < 0, <52 the road r->relt, —7t^t^Q,dl the
road t^re~l\ —K^t^Q. Then, Cauchy's theorem applied in the half-
plane c/(z) < <jf (resp. «/(z) > — ^) which is a star-shaped domain (9.7.1)
yields

f dz f dz f dz f dz

------=------- and -------= ------~.

Hence

r dz r dz f"

27ri(j(x1;y)-j(;c2;y))=--------- —— +

j52 z —• i^ j^j z -f zc Jo

Now the left-hand side is independent of £, and when ^ tends to 0, the right-
hand side tends to 2ni, using the fact that \y(t)\ ^ r for 0 < t^ a>, the mean
value theorem (to majorize the last integral), and (8.11.1).

(Ap.4.2.3) We now turn to the case in which H contains no segment with
distinct extremities. Let a, b be two distinct points of H, S the segment
of extremities a, b; we may again suppose that S is a closed interval in R.
By assumption, there is at least one point x e S n (C — H); let J be the
connected component of x in S n (C — H), which is an open interval ]y, z[
since S n (C — H) is open in R ((3.19.1) and (3.19.5)); moreover its extremities
y, z are in H. Let g be a homeomorphism of H onto the unit circle U, and
let g(y) = eic, g(z) = eid, where we may suppose that c < d < c + 2n (9.5.7).
Let U1? U2 be the simple arcs, images of 7->e**, c^t^d, and t-+elt,
d < t < c + 27i, and let Hl5 H2 be their images by the homeomorphism /
of U onto H, inverse to g. Using (9.5.7), we see immediately that there is ax2l'y) = ±1 for